The purpose of this research is to study models for the spatial spread of epidemics, using mathematical analysis and computational methods. In the main model, susceptibles, infectives and immunes (removals) are sited on a lattice; the probability of the infection of a susceptible is due to the number of infectives in its nearest neighborhood. Both discrete and continuous time Markov models on the lattice will be considered, and spatial data used to validate them. This validation will bc based on goodness of fit tests for Markov fields, some already known, and others to be developed. The second deterministic model considers the geographical spread of an infection by travellers. Here, numerical methods based on data for travel between major US centers will be used to estimate the spread of diseases such as measles and AIDS throughout the country. The research will also focus on how control measures such as screening, inoculation and quarantine may affect the transmission of a disease. These measures will be reflected in parameter changes in the models, but may also be studied by "Machine Learning" methods. The work will stress computational approaches to spatial epidemics, both in the visual validation of the progress of an epidemic, and the search for an optimal control strategy. The procedures developed will be simple enough to be used by medical practitioners and public health workers; they will thus gain insight into the geographical spread and control of epidemics.